Conformal Mappings and Loewner Evoluation

点击翻译按钮获取中文摘要

Proposal Number: DMS-0201435PI: Donald Marshall and Steffen RohdeABSTRACTConformal mapping and Loewner evolutionsMarshall and Rohde will investigate conformal mappings generated by the Loewner differential equation, and related topics. TheLoewner differential equation describes the flow associated withthe conformal mappings onto a continuously decreasing sequence ofsimply connected planar domains. It relates a sequence of domainsto a real-valued function, the driving term of the equation.Schramm's recent discovery of the stochastic Loewner evolutionSLE, the Loewner equation driven by one-dimensional Brownianmotion, has opened up a new area of investigations involvingconformal mappings, probability theory and mathematical physics.The Loewner equation is also related to an algorithm fornumerical conformal mapping.Conformal mappings have applications in many areas, both within and outside of mathematics, such as control theory, heatconduction, fluid dynamics, and complex dynamics. They are oftenused to change coordinates from one region to a simpler regionlike a disc. Regions with smooth boundaries are well understood.However, the appearance of fractals in many branches of science led to the natural problem of investigating regions bounded by highly nonsmooth, fractal curves, from the conformal mappingpoint of view. In recent years, fractal curves generated byrandom processes arising, for instance, in statistical physics,have received enormous attention. The core of Marshall's andRohde's research is to better understand random fractal curvesby means of conformal mappings, and conversely to study someproblems about conformal mappings by analyzing domains boundedby fractal curves.